Simultaneous confidence bounds with applications to drug stability studies

by Lu, Xiaojing.

John T. Chen, Advisor The focus of this research was to develop simultaneous confidence bounds for all contrasts of several regression lines with a constrained explanatory variable. The pioneering work of Spurrier provided a set of simultaneous confidence bounds for exact inference on all contrasts of several simple linear regression lines over the entire range (??, ?) using the same n design points. However, in many applications, the explanatory variables are constrained to smaller intervals than the entire range (??, ?). Spurrier clearly stated in the article (JASA, 1999) that the inference problem becomes much more complicated when the explanatory variable is bounded to a given interval. In fact, Wei Liu et al. (JASA, 2004) have investigated this issue, but were unable to solve the problem. Instead, they were obliged to rely on simulation based methods which produced approximate probability points for simultaneous comparisons. A noted criticism of their method is that the results are not exact and the simulations must be repeated for each application. In this research, a set of simultaneous confidence bounds for all contrasts of several linear regression lines was constructed for when the explanatory variable is restricted to a fixed interval, [?x0, x0], where x0 is a predetermined constant. These results greatly improve those of Spurrier since restricting the explanatory variable to a smaller interval results in narrower confidence bounds. Further, since the methods of this research are exact, they are superior to the earlier work of Wei Liu et al. A significant area of this research concerned a certain statistic that plays a crucial role in constructing confidence bounds with a constrained explanatory variable, and a pivotal iii quantity that aids in the discovery of critical values for determining the confidence bounds. The pivotal quantity is the maximum value of an associated function, and the statistic is the cut-off point at which the function is optimized. It is of primary importance to find a closed-form expression for the pivotal quantity and to derive its exact distribution. In this research, both of these problems were solved. In addition, the exact distribution of the statistic was found to be a standard Cauchy distribution; in fact, amazingly, it has also been shown that the statistic is independent of the pivotal quantity. These research results shed surprising new light on long standing knotty problems in biostatistics. Applications of this method to drug stability studies were examined. In situations where multiple batches of a drug product are manufactured, it is desired to pool data from different batches to obtain a single shelf-life for all batches. This research provided a new pooling method that was demonstrated to be more versatile and efficient than the existing pooling procedures. iv